#7622878

order	512
nilpotency class	4
isoclinism	isoclinic class of #7622875
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁰

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	C₂²×B(4,𝔽₂)
commutator, Frattini, cosocle	C₂×D₄	C₂⁵
LCS	G ▷ C₂×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂×D₄ ▷ C₂² ▷ C₂ ▷ C₁

3792 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 5 generators on 32 points:

(1,2)(3,9)(4,10)(5,11)(6,12)(7,13)(8,15)(14,24)(16,25)(17,26)(18,27)(19,29)(20,23)(21,30)(22,31)(28,32);(1,3)(2,8)(4,14)(5,17)(6,15)(7,16)(9,12)(10,22)(11,28)(13,23)(18,31)(19,32)(20,21)(24,27)(25,30)(26,29);(1,3)(2,9)(4,17)(5,14)(6,20)(7,16)(8,12)(10,26)(11,24)(13,25)(15,21)(18,31)(19,32)(22,29)(23,30)(27,28);(1,4,7,5)(2,10,13,11)(3,14,16,17)(6,18,21,19)(8,22,23,28)(9,24,25,26)(12,27,30,29)(15,31,20,32);(1,5,7,4)(2,11,13,10)(3,14,16,17)(6,19,21,18)(8,22,23,28)(9,24,25,26)(12,29,30,27)(15,31,20,32)

Polycyclic presentation:

2^9|d^2=i,e^2=i,b^a=bf,c^a=cg,h^a=hi,e^b=ei,g^b=ghi,d^c=di,f^c=fh,g^f=gi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72234209, #72234211, #72249654, …
maximal normal subgroup of	#72234209, #72234211, #72249654, …
maximal quotient of	#72251874, #72282033, #72312502, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7530074);